Non-Binary Matroids Having Four Non-Binary Elements

An element \(e\) of a matroid \(M\) is called non-binary when \(M\backslash e\) and \(M/e\) are both non-binary matroids. Oxley in \({6}\) gave a characterization of the \(3\)-connected non-binary matroids without non-binary elements. In {4}, we constructed all the \(3\)-connected matroids having exactly \(1\), \(2\) or \(3\) non-binary elements. In this paper, we will give a characterization of the \(3\)-connected matroids having exactly four non-binary elements.